Some Inversion Formulas for the Cone Transform

TL;DR

This paper develops inversion formulas for the cone transform, a Radon-type transform over conical surfaces, with applications in medical imaging, astronomy, and security, focusing on overdetermined data scenarios.

Contribution

It introduces integral relations between cone and Radon transforms in ^n and derives inversion formulas for the most general cone transform, emphasizing overdetermined data.

Findings

Derived integral relations between cone and Radon transforms.

Established inversion formulas for the general cone transform.

Focused on formulas suitable for overdetermined data in practical applications.

Abstract

Several novel imaging applications have lead recently to a variety of Radon type transforms, where integration is done over a family of conical surfaces. We call them \emph{cone transforms} (in 2D they are also called \emph{V-line} or \emph{broken ray} transforms). Most prominently, they are present in the so called Compton camera imaging that arises in medical diagnostics, astronomy, and lately in homeland security applications. Several specific incarnations of the cone transform have been considered separately. In this paper, we address the most general (and overdetermined) cone transform, obtain integral relations between cone and Radon transforms in $\mathbb{R}^n$, and a variety of inversion formulas. In many applications (e.g., in homeland security), the signal to noise ratio is very low. So, if overdetermined data is collected (as in the case of Compton imaging), attempts to reduce the dimensionality might lead to essential elimination of the signal. Thus, our main concentration is on obtaining formulas involving overdetermined data.